let GX be non empty TopSpace; :: thesis: for B1, B2, V being Subset of GX holds Down ((B1 /\ B2),V) = (Down (B1,V)) /\ (Down (B2,V))
let B1, B2, V be Subset of GX; :: thesis: Down ((B1 /\ B2),V) = (Down (B1,V)) /\ (Down (B2,V))
Down ((B1 /\ B2),V) = (B1 /\ B2) /\ V by CONNSP_3:def 5;
then A1: Down ((B1 /\ B2),V) = B1 /\ (B2 /\ (V /\ V)) by XBOOLE_1:16
.= B1 /\ ((B2 /\ V) /\ V) by XBOOLE_1:16
.= (B1 /\ V) /\ (B2 /\ V) by XBOOLE_1:16 ;
Down (B1,V) = B1 /\ V by CONNSP_3:def 5;
hence Down ((B1 /\ B2),V) = (Down (B1,V)) /\ (Down (B2,V)) by A1, CONNSP_3:def 5; :: thesis: verum